# A Coupled Seepage–Deformation Model for Simulating the Effect of Fracture Seepage on Rock Slope Stability Using the Numerical Manifold Method

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## Abstract

**:**

## 1. Introduction

## 2. Coupled Seepage–Deformation Model for the NMM

#### 2.1. Fundamentals of the NMM

#### 2.2. Basic Formulas of Seepage–Deformation Coupling Model

- The rock matrix is impermeable, and only fracture seepage is considered.
- The fluid flow is assumed to be laminar, steady, viscous, and incompressible, and the fluid pressure is always positive.
- The fluid flow has no effect on the strength of the material, and no hydraulic fractures initialize and propagate in the model.

_{i}in Figure 3); then, the block system, consisting of a fracture network and seepage paths, is established. In this way, the coupling between the fracture flow and deformation block is built. For example, the aperture between two connected blocks, denoted by ③ and ⑥, is filled with flowing fluid. The seepage path of 1–6 is divided into five small line segments by mathematical meshing, i.e., 1–2, 2–3, 3–4, 4–5, and 5–6, and each divided seepage path has two surfaces, which constitutes a fracture seepage element. Each coupled surface and separated seepage path is exactly a line segment of a loop belonging to blocks ③ and ⑥, respectively. Therefore, the closed loops are not only used to compute the contact forces but are also employed to generate the potential fracture network for calculating fluid flow. The simulations of seepage pressure in this study are based on the fracture coupling model, and the triangle meshing technique is employed to form MEs (e.g., ${E}_{i}$ in Figure 3) in the computations.

**n**, and ${a}_{k+1}$ is the relative distance between points k + 1 in the normal direction, as shown in Figure 5.

#### 2.3. Fluid Pressure on Manifold Element Boundaries

## 3. Modeling of Seepage along Fractured Network by the NMM

#### 3.1. Calculation of Fluid Pressure along the Fracture Network

#### 3.2. Calculation of Factor of Safety (FoS) along Rock Fractures

## 4. Numerical Examples

#### 4.1. Verification of the Coupled Seepage–Deformation Model

_{1}and h

_{2}. The bottom boundary of the model is assumed to be impermeable (i.e., q = 0). The analytical solution of the total discharge for this model is [47]:

^{3}, Young’s modulus is 75 GPa, Poisson’s ratio is 0.22, fluid viscosity coefficient is $1.005\times {10}^{-3}$ Pa s, cohesion of the fracture is 0.1 MPa/m

^{2}, friction angle is 25°, and the normal contact stiffness is 200 GPa. To obtain computational accuracy, a constant time step, $\u2206t$, 5 × 10

^{−8}s is applied. In the model, a total of 2800 PCs, 3688 MEs, and 128 flow segments are obtained, and 7 measured points are prescribed to sum the discharge of the fracture network system. The total discharge can then be written as:

#### 4.2. Fracture Seepage of Fluid Flow through a Regular Fracture Network

_{h}= 0.2 m are assembled (as seen in Figure 10). To simulate the pressure head distribution of the phreatic surface along the bottom of impermeable boundaries, the fractures are prescribed a uniform aperture, a, of 0.4 mm; the hydraulic boundary conditions used in the NMM are listed as follows: a constant water head, h, of 2.9 m is assigned to the left side of the model, a constant velocity of 50 cm/s at the lower-right corner is prescribed, and there is no flow rate for the other boundaries.

- h0 = 0.8 m is prescribed at the left side of the model, and except for the impermeable base, the other boundaries are free surfaces;
- h0 = 0.8 m is prescribed at the left side of the model, but the right side has a water head of h = 0.4 m.

#### 4.3. Joint Seepage in an Arbitrary Complex Rock Fracture Network

^{3}, Young’s Modulus is 3.1 GPa, Poisson’s ratio is 0.35, and the kinematic viscosity of water is 1.005 × 10

^{−3}Pa s. Four representative tests (T1–7 to T1–10) are analyzed by the NMM, and the simulated results are listed in Table 2, in which the errors in head value, E

_{r}, at each measured port are expressed as:

^{−14}m

^{2}) by FEM code indicates the pressure distribution when porous media is considered (Figure 14a), but the simulated results using the proposed model (Figure 15b) by the NMM agree very well with both the hydraulic pressures and the flow rates at the measured ports. It is clear that the proposed seepage–deformation model is more efficient and flexible for simulating steady-state fluid flow in discrete fracture networks.

#### 4.4. Rock Slope Stability Analysis Using the Coupled Seepage–Deformation Model

- The effect of groundwater is ignored, and only the self-weight of the rock mass is considered (i.e., Condition 1);
- The effect of groundwater is not ignored, the highest groundwater head is located at the top of the slope (see Figure 15), and the bottom of the slope is taken as an outlet for the water flow (i.e., Condition 2);
- The highest groundwater head is the same as Condition 2, but considering the condition under a frozen state of the rock slope surface, the groundwater outlet is blocked (i.e., Condition 3).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**Description of hydraulic aperture with average normal displacement: (

**a**) non-parallel fracture element; (

**b**) ideal parallel fracture element.

**Figure 13.**Experiment model in [49]: (

**a**) schematic of the complete laboratory device; (

**b**) fracture flow network in the physical model.

**Figure 14.**Seepage modeling using the numerical methods (e.g., T1–10): (

**a**) hydraulic pressure with rock matrix permeability (e.g., 1.0 × 10

^{−14}m

^{2}) by the FEM code [49]; (

**b**) flow rate without rock matrix permeability by the NMM.

**Figure 15.**The NMM model of the rock slope with fractures: (

**a**) block cutting of the model; (

**b**) fracture networks; (

**c**) NMM meshing of the model.

**Figure 17.**Fluid flow and pressure head (unit: mm) when groundwater is considered: (

**a**) Condition 2; (

**b**) Condition 3.

**Figure 18.**Simulated failure process of the rock slope (unit: m): (

**a**) step 0; (

**b**) step 620; (

**c**) step 1527; (

**d**) step 2866; (

**e**) step 5161; (

**f**) step 7015.

Work No. | T1–7 | T1–8 | T1–9 | T1–10 |
---|---|---|---|---|

$H$ (cm) | 115.70 | 134.37 | 151.38 | 227.20 |

$h$ (cm) | 71.37 | 71.37 | 72.01 | 54.36 |

$\u2206h$ (cm) | 44.33 | 63.00 | 79.37 | 172.84 |

$i\left(\u2206h/L\right)$ | 0.727 | 1.033 | 1.302 | 2.835 |

**Table 2.**Comparison of the hydraulic heads between the coupled model and experimental results in [49].

Port No. | T1–7 | T1–8 | T1–9 | T1–10 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{h}}_{\mathit{n}}\left(\mathbf{cm}\right)$ | ${\mathit{h}}_{\mathit{l}}\left(\mathbf{cm}\right)$ | ${\mathit{E}}_{\mathit{r}}(\%)$ | ${\mathit{h}}_{\mathit{n}}\left(\mathbf{cm}\right)$ | ${\mathit{h}}_{\mathit{l}}\left(\mathbf{cm}\right)$ | ${\mathit{E}}_{\mathit{r}}(\%)$ | ${\mathit{h}}_{\mathit{n}}\left(\mathbf{cm}\right)$ | ${\mathit{h}}_{\mathit{l}}\left(\mathbf{cm}\right)$ | ${\mathit{E}}_{\mathit{r}}(\%)$ | ${\mathit{h}}_{\mathit{n}}\left(\mathbf{cm}\right)$ | ${\mathit{h}}_{\mathit{l}}\left(\mathbf{cm}\right)$ | ${\mathit{E}}_{\mathit{r}}(\%)$ | |

1 | 112.74 | 112.65 | −0.20 | 130.06 | 130.56 | 0.79 | 146.18 | 147.45 | 1.60 | 215.36 | 217.81 | 1.42 |

2 | 113.06 | 113.16 | 0.23 | 129.97 | 131.19 | 1.94 | 146.63 | 147.70 | 1.35 | 217.03 | 218.57 | 0.89 |

3 | 109.44 | 109.22 | −0.50 | 124.51 | 126.37 | 2.95 | 140.17 | 142.49 | 2.92 | 201.45 | 201.93 | 0.28 |

4 | 112.95 | 112.52 | −0.97 | 130.32 | 130.56 | 0.38 | 146.53 | 146.94 | 0.52 | 216.78 | 217.04 | 0.15 |

5 | 113.28 | 113.54 | 0.59 | 130.78 | 131.32 | 0.86 | 147.16 | 148.08 | 1.16 | 217.15 | 218.06 | 0.53 |

6 | 114.36 | 114.68 | 0.72 | 133.19 | 133.10 | −0.14 | 150.27 | 150.11 | −0.20 | 223.86 | 224.03 | 0.10 |

7 | 107.89 | 108.20 | 0.70 | 123.58 | 124.84 | 2.00 | 138.22 | 140.97 | 3.46 | 193.95 | 200.79 | 3.96 |

8 | 108.12 | 108.46 | 0.77 | 124.26 | 125.10 | 1.33 | 138.49 | 141.35 | 3.60 | 194.88 | 201.17 | 3.64 |

9 | 90.27 | 91.19 | 2.08 | 102.43 | 102.11 | −0.51 | 111.75 | 114.81 | 3.86 | 141.72 | 141.61 | −0.06 |

10 | 88.32 | 89.41 | 2.46 | 97.83 | 98.55 | 1.14 | 105.38 | 111.00 | 7.08 | 134.37 | 133.86 | −0.30 |

11 | 91.95 | 91.82 | −0.29 | 101.35 | 103.51 | 3.43 | 110.73 | 116.33 | 7.06 | 139.26 | 144.40 | 2.97 |

12 | 97.68 | 97.79 | 0.25 | 109.37 | 111.38 | 3.19 | 121.64 | 125.60 | 4.99 | 154.32 | 165.10 | 6.24 |

13 | 98.48 | 98.68 | 0.45 | 110.69 | 112.40 | 2.71 | 124.38 | 127.00 | 3.30 | 156.87 | 167.39 | 6.09 |

14 | 95.88 | 95.63 | −0.56 | 107.41 | 107.32 | −0.14 | 119.59 | 120.90 | 1.65 | 148.37 | 157.99 | 5.57 |

15 | 97.06 | 96.52 | −1.22 | 108.14 | 109.73 | 2.52 | 120.15 | 123.44 | 4.15 | 152.64 | 163.20 | 6.11 |

16 | 96.97 | 97.28 | 0.70 | 108.66 | 110.36 | 2.70 | 120.63 | 123.95 | 4.18 | 153.33 | 163.96 | 6.15 |

17 | 76.75 | 77.72 | 2.19 | 81.06 | 82.30 | 1.97 | 83.47 | 89.41 | 7.48 | 81.06 | 86.49 | 3.14 |

18 | 75.22 | 76.45 | 2.77 | 77.98 | 78.99 | 1.60 | 81.46 | 85.09 | 4.57 | 75.12 | 77.98 | 1.65 |

19 | 80.17 | 81.79 | 3.65 | 89.16 | 88.65 | −0.81 | 90.16 | 96.14 | 7.53 | 94.75 | 100.46 | 3.30 |

20 | 77.28 | 78.74 | 3.29 | 82.57 | 83.31 | 1.17 | 91.07 | 90.68 | −0.49 | 83.16 | 89.66 | 3.76 |

21 | 83.43 | 84.33 | 2.03 | 93.08 | 92.96 | −0.19 | 97.32 | 100.46 | 3.96 | 105.47 | 113.67 | 4.74 |

22 | 76.45 | 77.34 | 2.01 | 83.78 | 83.06 | −1.14 | 87.95 | 87.63 | −0.40 | 76.04 | 87.76 | 6.78 |

23 | 75.56 | 75.82 | 0.59 | 78.64 | 79.50 | 1.37 | 81.26 | 83.82 | 3.23 | 70.73 | 78.11 | 4.27 |

24 | 74.34 | 74.17 | −0.38 | 76.86 | 77.09 | 0.37 | 79.34 | 80.39 | 1.32 | 66.18 | 71.63 | 3.15 |

Material | Young’s Modulus (GPa) | Poisson’s Ratio | Tensile Strength (MPa) | Cohesion (kPa) | Internal Friction Angle (°) | Density (kN/m^{3}) |
---|---|---|---|---|---|---|

Rock mass | 5 | 0.25 | 1.0 | 1.5 | 45 | 24 |

Tunnel lining | 20 | 0.2 | 1.5 | 2.0 | 45 | 25 |

Fracture | / | / | / | 0.0 | 30 | / |

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## Share and Cite

**MDPI and ACS Style**

Qu, X.; Zhang, Y.; Chen, Y.; Chen, Y.; Qi, C.; Pasternak, E.; Dyskin, A.
A Coupled Seepage–Deformation Model for Simulating the Effect of Fracture Seepage on Rock Slope Stability Using the Numerical Manifold Method. *Water* **2023**, *15*, 1163.
https://doi.org/10.3390/w15061163

**AMA Style**

Qu X, Zhang Y, Chen Y, Chen Y, Qi C, Pasternak E, Dyskin A.
A Coupled Seepage–Deformation Model for Simulating the Effect of Fracture Seepage on Rock Slope Stability Using the Numerical Manifold Method. *Water*. 2023; 15(6):1163.
https://doi.org/10.3390/w15061163

**Chicago/Turabian Style**

Qu, Xiaolei, Yunkai Zhang, Youran Chen, Youyang Chen, Chengzhi Qi, Elena Pasternak, and Arcady Dyskin.
2023. "A Coupled Seepage–Deformation Model for Simulating the Effect of Fracture Seepage on Rock Slope Stability Using the Numerical Manifold Method" *Water* 15, no. 6: 1163.
https://doi.org/10.3390/w15061163